Place the eggs into a saucepan and fill with water to cover the eggs by 1 inch. Cover the saucepan and bring the water to a boil over high heat. Once the water is boiling, remove from the heat and let the eggs stand in the hot water for 15 minutes. Pour out the hot water, then cool the eggs under cold running water in the sink. Peel once cold and cut into slices; set aside.
Meanwhile, place a steamer insert into a saucepan, and fill with water to just below the bottom of the steamer. Cover, and bring the water to a boil over high heat. Add the asparagus, recover, and steam until just tender, 2 to 6 minutes depending on thickness.
While the asparagus is steaming, whisk together the vinegar, mustard, salt, pepper, green bell pepper, relish, parsley, and chives in a large mixing bowl. Slowly pour in the olive oil while whisking quickly to create a dressing. Toss the hot, steamed asparagus in the dressing, then place into the refrigerator for at least an hour.
To serve, divide the baby greens onto individual plates. Divide the asparagus spears among the salads, then garnish with radish slices, egg slices, and quartered tomatoes.